Length of Basis Doesn't Depend on Basis

Any two basis of finite-dimensional vector space have the same length.
constituents A finite-dimensional vector space V Basis B_1, B_2 be bases in V requirements Given B_1, B_2 are basis in V, we know that they are both linearly independent and spans V. We have that the length of linearly-independent list \leq length of spanning list.
Let’s take first B_1 as linearly independent and B_2 as spanning:
We have then len(B_1) \leq len(B_2)
Swapping roles:
We have then len(B_2) \leq len(B_1)
As both of this conditions are true, we have that len(B_1)=len(B_{2}). \blacksquare